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 A334796 a(n) = (A070939(A334769(n)) - A334770(n))/3. 5
 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 4, 4, 4, 4, 2, 3, 3, 4, 4, 3, 3, 4, 4, 3, 5, 5, 5, 5, 3, 2, 4, 4, 4, 4, 2, 3, 5, 5, 5, 5, 5, 3, 5, 2, 5, 4, 5, 4, 4, 5, 4, 5, 2, 5, 3, 5, 6, 6, 6, 6, 3, 4, 5, 5, 4, 3, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 3, 4, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(m) applied recursively until we reach a single bit. A334556 is the sequence of rotationally symmetrical T(m). A central zero-triangle (CZT) is a field of contiguous 0-bits, listed in A334769, a subset of A334556. CZTs have side length k = A334770(n), surrounded on all sides by a layer of 1 bits, and generally j > 1 bits of any parity. This sequence describes the "frame width" j. Smallest n with a given value of j appears in A334836. - Michael De Vlieger, May 20 2020 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..10000 Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020. Michael De Vlieger, Diagram montage showing XOR-triangles for terms in certain linear recurrences and their bit-reversals, illustrating relations in their appearance, most significantly, constant frame width. Michael De Vlieger, Diagram montage showing the first dozen XOR-triangles exhibiting frame widths of 2, 3, 4, ..., 12 by row. EXAMPLE a(4) pertains to T(599), with A334770(4) = 4. (1 + A070939(599) - 4)/3 = (1 + 9 - 4)/3 = 6/3 = 2, thus a(4) = 2. (Diagram, replacing 0 with “.”):   1 . . 1 . 1 . 1 1 1    1 . 1 1 1 1 1 . .     1 1 . . . . 1 .      . 1 . . . 1 1       1 1 . . 1 .        . 1 . 1 1         1 1 1 .          . . 1           . 1            1 a(11) pertains to T(2359), with A334770(11) = 3. (1 + A070939(2359) - 4)/3 = (1 + 11 - 3)/3 = 9/3 = 3, thus a(11) = 3. (Diagram):   1 . . 1 . . 1 1 . 1 1 1    1 . 1 1 . 1 . 1 1 . .     1 1 . 1 1 1 1 . 1 .      . 1 1 . . . 1 1 1       1 . 1 . . 1 . .        1 1 1 . 1 1 .         . . 1 1 . 1          . 1 . 1 1           1 1 1 .            . . 1             . 1              1 From Michael De Vlieger, May 14 2020: (Start) Linear recurrences that produce XOR-triangles with frame length j (table may not be exhaustive): j   LR          Lower               Upper ----------------------------------------------------- 2   (5, -4)     {39, 151}           {57, 223} 3   (17, -16)   {543, 8607}         {993, 15969}                 {1379, 22115}       {1589, 25397}                 {1483, 23755}       {1693, 27037}                 {2359, 37687}       {3785, 60617} 4   (17, -16)   {22243, 356067}     {25525, 408501}                 {39047, 624775}     {57625, 921881}                 {40679, 650983}     {59257, 948089}                 {171475, 2743763}   {208613, 3337957}                 {356067, 5697251}   {408501, 6536117} ... (End) MATHEMATICA Block[{f, s = Rest[Import["https://oeis.org/A334556/b334556.txt", "Data"][[All, -1]] ]}, f[n_] := NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]; Array[Block[{n = s[[#]]}, If[# == 0, Nothing, (1 + Floor@ Log2[n] - #)/3] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, f[n][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]] ], r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Length@ s - 1, 2] ] CROSSREFS Cf. A038554, A070939, A334556, A334769, A334770, A334836. Sequence in context: A084126 A136032 A135975 * A140361 A237769 A187182 Adjacent sequences:  A334793 A334794 A334795 * A334797 A334798 A334799 KEYWORD nonn AUTHOR Michael De Vlieger, May 12 2020 STATUS approved

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Last modified September 23 16:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)